If you want a zillion, do what Fred suggested.
If you want a few, draw your object and select it. From there you can go to OBJECT, TRANSFORM, MOVE. From there you can enter in distances and angles for a movement OR a copy. One the object has been copied once, OBJECT, TRANSFORM AGAIN (Command-D on Mac, guessing CTRL-D on PC) will continue to copy (or move, depending on which you had originally selected) for as many times as you continue.
Once you have a bunch of your object going in one direction, you could select them all and then go back to OBJECT, TRANSFORM, MOVE and specify a different direction and hit COPY. You can then hit Command-D/CTRL-D until satisfied.
This probably isn't the best approach for 1000 units but it works well enough for smaller numbers.
If you do step and repeat stuff all of the time in Illustrator, I am sure there are both commercial and freeware plug-ins.